In this paper, we used principal component analysis (PCA) [1]} in order to find a modal basis for pivoted QR factorization and to identify a low-dimensional representation of the state and its covariance for determinantal D-optimal selection techniques on the shock-mixing layer flow.
In order to perform PCA, one needs an appropriate inner product on the space in which the data lives.
In the case of the shock-mixing layer problem, we use the energy-based inner product for compressible flows developed in [2]} together with trapezoidal quadrature weights to approximate the integrals of the spatial fields over a stretched computational grid.
In this problem, the data consists of vectors \({\mathbf {z}}\)  whose elements are the streamwise velocity \(u\) , transverse velocity \(v\) , and the local speed of sound \(a\)  over a \(321\times 81\)  computational grid.
The inner product between two snapshots \({\mathbf {z}}\)  and \({\mathbf {z}}^{\prime }\)  is defined by
\(\left\langle {\mathbf {z}},\ {\mathbf {z}}^{\prime } \right\rangle = {\mathbf {z}}^T{\mathbf {W}}{\mathbf {z}}^{\prime } = \sum _{i=1}^{321}\sum _{j=1}^{81}w_{i,j}\left( u_{i,j}^2 + v_{i,j}^2 + a_{i,j}^2 \right) \\\approx \int _{\Omega }\left[u(\xi _1,\xi _2)^2 + v(\xi _1,\xi _2)^2 + a(\xi _1,\xi _2)^2 \right] d\xi _1 d\xi _2,\) 
